Question

Using vector method, find the incentre of the triangle whose vertices are $P(0,4,0),Q(0,0,3)$ and $R(0,4,3)$

Answer 1

$\overrightarrow{P}=4\hat{j}$

$\overrightarrow{Q}=3\hat{k}$

$\overrightarrow{R}=4\hat{j}+3\hat{k}$

$\overrightarrow{PQ}=\overrightarrow{Q}-\overrightarrow{P}=-4\hat{j}+3\hat{k}$

$\overrightarrow{QR}=\overrightarrow{R}-\overrightarrow{Q}=4\hat{j}$

$\overrightarrow{R}P=\overrightarrow{P}-\overrightarrow{R}=-3\hat{k}$

$|\overrightarrow{PQ}|=5,|\overrightarrow{QR}|=4,|\overrightarrow{RP}|=3$

The incenter is the intersection of three angle bisectors,

$\therefore \overrightarrow{I}=\frac{|\overrightarrow{QR}|\cdot \overrightarrow{P}+|\overrightarrow{RP}|\cdot \overrightarrow{Q}+|\overrightarrow{PQ}|\cdot \overrightarrow{R}}{|\overrightarrow{PQ}|+|\overrightarrow{QR}|+|\overrightarrow{RP}|}$

$\Rightarrow \overrightarrow{I}=\frac{16\hat{j}+9\hat{k}+20\hat{j}+15\hat{k}}{12}$

$\Rightarrow \overrightarrow{I}=3\hat{j}+2\hat{k}$

$\therefore ,$ Coordinates of incentre are $(0,3,2)$